Transcript introduction to statistics and statistical inference

STATISTICAL TOOLS
NEEDED
IN
ANALYZING TEST RESULTS
Prof. Yonardo Agustin Gabuyo
Statistics is a branch of science
which deals with the collection,
presentation, analysis and
interpretation of quantitative
data.
Branches of Statistics
Descriptive statistics

methods
concerned
w/
collecting,
describing,
and
analyzing a set of data without
drawing
conclusions
(or
inferences) about a large group
Inferential statistics
 methods concerned with
the analysis of a subset of
data leading to predictions
or inferences about the
entire set of data or
population.
Examples of Descriptive Statistics
 Presenting the Philippine population
by constructing a graph indicating the
total number of Filipinos counted during
the last census by age group and sex
 The Department of Social Welfare and
Development (DSWD) cited statistics
showing an increase in the number of
child abuse cases during the past five
years.
Examples of Inferential Statistics
Source: Pilot Training Course on Teaching Basic Statistics by Statistical Research and Training Center Philippine Statistical
Association , Inc.
A new milk formulation designed to improve the
psychomotor development of infants was tested on
randomly selected infants. Based on the results, it
was concluded that the new milk formulation is
effective
in
improving
the
psychomotor
development of infants.
Example
Teacher Ron-nick gave a personality
test measuring shyness to 25,000
students. What is the average
degree of shyness and what is the
degree to which the students differ
in shyness are the concerns of
_________ statistics.
A. inferential
B. graphic
C. correlational
D. descriptive
Example
This is a type of statistics that give/s
information about the sample
being studied.
a. Inferential and co-relational
b. Inferential
c. Descriptive
d. Co relational
Inferential Statistics
Source: Pilot Training Course on Teaching Basic Statistics by Statistical Research and Training Center Philippine
Statistical Association , Inc.
Larger Set
(N units/observations)
Smaller Set
(n units/observations)
Inferences and
Generalizations
Types of Variables
VARIABLES
Qualitative
Discrete
Quantitative
Continuous
Qualitative variables
 variables that can be express in
terms of properties, characteristics,
or classification(non-numerical
values).
Quantitative Variables
 variables that can be express in terms
of numerical values.
a)Discrete- variables that can be express
in terms of whole number.
b)Continuous- variables that can be
express in terms whole number, fraction
or decimal number.
Levels of Measurement
1. Nominal
 Numbers or symbols used to
classify
2. Ordinal scale
 Accounts for order; no indication
of distance between positions
3. Interval scale
 Equal intervals; no absolute zero
4. Ratio scale
 Has absolute zero
Methods of Collecting Data
 Objective
Subjective
Method
Method
Use
of Existing Records
Methods of Presenting Data
 Textual
 Tabular
 Graphical
Summary Measures
Location
Variation
Skewness
Kurtosis
Percentile
Maximum
Minimum
Quartile
Decile
Range
Variance
Central
Tendency
Mean
Inter-quartile
Range
Mode
Median
Coefficient of
Variation
Standard Deviation
Measures of Location
A Measure of Location summarizes a data
set by giving a “typical value” within the
range of the data values that describes its
location relative to entire data set.
Some Common Measures:
Minimum, Maximum
Central Tendency
Percentiles, Deciles, Quartiles
Maximum
and
Minimum
 Minimum is the smallest value in the data set,
denoted as MIN.

Maximum is the largest value in the data set,
denoted as MAX.
Measure of Central Tendency
 A single value that is used to
identify the “center” of the data
it is thought of as a typical value
of the distribution
precise yet simple
most representative value of the
data
Mean
 Most common measure of the center
 Also known as arithmetic average
Population Mean
Sample Mean
Properties of the Mean
 may not be an actual
observation in the data set.
 can be applied in at least
interval level.
 easy to compute.
 every observation contributes to
the value of the mean.
Properties of the Mean
subgroup means can be combined
to come up with a group mean
 easily affected by extreme values
0 1 2 3 4 5 6 7 8 9 10
Mean = 5
0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 6
Median
 Divides the observations into two equal
parts.
 If n is odd, the median is the middle
number.
 If n is even, the median is the average of
the 2 middle numbers.
 Sample median denoted as
while population median is denoted as
~
x
~

Properties of a Median
 may not be an actual observation in the data
set
 can be applied in at least ordinal level
 a positional measure; not affected by
extreme values
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5
Mode
 the score/s that occurs most
frequently
 nominal average
 computation of the mode for
ungrouped or raw data
0 1 2 3 4 5 6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
No Mode
Properties of a Mode
 can be used for qualitative as well
as quantitative data
 may not be unique
 not affected by extreme values
 may not exist
Mean, Median & Mode
Use the mean when:
 sampling stability is desired
 other measures are to be
computed
Mean, Median & Mode
Use the median when:
 the exact midpoint of the
distribution is desired
 there are extreme observations
Mean, Median & Mode
Use the mode when:
 when the "typical" value is desired
 when the dataset is measured on
a nominal scale
Example
Which measure(s) of central
tendency is(are) most appropriate
when the score distribution is
skewed?
A. Mode
B. Mean and mode
C. Median
D. Mean
Example
In one hundred-item test, what does
Jay-R’s score of 70 mean?
A. He surpassed 70 of his classmate
in terms of score
B. He surpassed 30 of his classmates
in terms of score
C. He got a score above mean
D. He got 70 items correct
Example
Which of the following measures is
more affected by an extreme score?
A. Semi- inter quartile range
B. Median
C. Mode
D. Mean
Example
The sum of all the scores in a
distribution always equals
a. The mean times the interval size
b. The mean divided by the interval
size
c. The mean times N
d. The mean divided by N
Example
Teacher B is researching on family
income distribution which is
symmetrical. Which measure/s of
central tendency will be most
informative and appropriate?
A. Mode
B. Mean
C. Median
D. Mean and Median
Example
What measure/s of central tendency
does the number 16 represent in
the following score distribution?
14,15,17,16,19,20,16,14,16?
a. Mode only
b. Mode and median
c. Median only
d. Mean and mode
Example
What is the mean of this score
distribution: 40, 42, 45, 48, 50, 52,
54, 68?
a. 51.88
b. 50.88
c. 49.88
d. 68
Example
Which is the correct about
MEDIAN?
a. It is measure of variability
b. It is the most stable measure of
central tendency
c. It is the 50th percentile
d. It is significantly affected by
extreme values
Example
Which measure(s) of central
tendency can be determined by
mere inspection?
a. Median
b. Mode
c. Mean
d. Mode and Median
Example
Here is a score distribution:
98,93,93,93,90,88,88,85,85,85,86,
70,70,51,34,34,34,, 20,18,15,12,9,8,3,1.
Which is a characteristics of the
scores distribution?
A. Bi-modal
B. Tri-modal
C. Skewed to the right D. No
discernible pattern
Example
Which is true of a bimodal score
distribution?
a. the group tested has two identical
scores that appeared most.
b. the scores are either high or low.
c. the scores are high.
d. the scores are low.
Example
STUDY THE TABLE THEN ANSWER THE QUESTION:
Scores
0-59
60-69
70-79
80-89
90-100
Percent of Students
2%
8%
39%
38%
13%
In which scores interval is the
median?
a. In the interval 80 to 89
b. In between the intervals of 60-69
and 70-79
c. In the interval 70-79
d. In the interval 60-69
How many percent of the students
got a score below 70?
a. 2%
b. 8%
c. 10%
d. 39%
Percentiles
 Numerical measures that
give the relative position of a
data value relative to the
entire data set.
 Percentage of the students in
the reference group who fall
below student’s raw score.
Divides the scores in the
distribution into 100 equal
parts (raw data arranged in
increasing or decreasing order
of magnitude).
 The jth percentile, denoted as
Pj, is the data value in the data
set that separates the bottom
j% of the data from the top
(100-j)%.
EXAMPLE
Suppose JM was told that relative to the
other scores on a certain test, his score was
the 97th percentile.
 This means that 97% of those who took
the test had scores less than JM’s score,
while 3% had scores higher than JM’s.
Deciles
Divides
the
scores
in
the
distribution into ten equal parts,
each part having ten percent of the
distribution of the data values below
the indicated decile.
 The 1st decile is the 10th percentile;
the 2nd decile is the 20th
percentile…..
 9th decile is the 90th percentile.
Quartiles
 Divides the scores in the distribution
into four equal parts, each part having
25% of the scores in the distribution of
the data values below the indicated
quartile.
 The 1st quartile is the 25th percentile; the
2nd quartile is the 50th percentile, also the
median and the 3rd quartile is the 75th
percentile.
Example
Robert Joseph’s raw score in the mathematics
class is 45 which equal to 96th percentile.
What does this mean?
a. 96% of Robert Joseph’s classmates got a
score higher than 45.
b. 96% of Robert Joseph’s classmates got a
score lower than 45.
c. Robert Joseph’s score is less than 45% of his
classmates.
d. Roberts Joseph’s is higher than 96% of his
classmates.
Example
Which one describes the percentile rank of a
given score?
a. The percent of cases of a distribution below
and above a given score.
b. The percent of cases of a distribution
below the given score.
c. The percent of cases of a distribution above
the given score.
d. The percent of cases of a distribution
within the given score.
Example
Biboy obtained a score of 85 in
Mathematics multiple choice tests.
What does this mean?
a. He has a rating of 85
b. He answered 85 items in the test correctly
c. He answered 85% of the test item correctly
d. His performance is 15% better than the
group
Example
Median is the 50th percentile as Q3 is to
a. 45th percentile
b. 70th percentile
c. 75th percentile
d. 25th percentile
Example
Karl Vince obtained a NEAT percentile
rank of 95. This means that
a. They have a zero reference point
b. They have scales of equal units
c. They indicate an individual’s relative
standing in a group
d. They indicate specific points in the
normal curve
Example
Markie obtained a NEAT percentile rank
of 95.
This means that
a. He got a score of 95.
b. He answered 95 items correctly.
c. He surpassed in performance of 95%
of his fellow examinees.
d. He surpassed in performance 0f 5% of
his fellow examinees.
Example
What is/are important to state when
explaining percentile-ranked tests to
parents?
I. What group took the test
II. That the scores show how students performed
in relation to other students
III. That the scores show how students
performed in relation to an absolute measure
A. II only B. I & III C. I & II D. III only
Measures of Variation
A measure of variation is a
single value that is used to
describe the spread of the
distribution.
 A measure of central tendency
alone does not uniquely
describe a distribution.
A look at dispersion… Pilot
Source: Training Course on Teaching Basic Statistics by Statistical Research and Training Center Philippine
Statistical Association , Inc.
Section A
11
12
13
14
15
16
17
18
19
20 21
Mean = 15.5
s = 3.338
20 21
Mean = 15.5
s = .9258
Section B
11
12
13
14
15
16
17
18
19
Section C
11
12
13
14
Mean = 15.5
s = 4.57
15
16
17
18
19
20 21
Two Types of Measures of Dispersion
Absolute Measures of Dispersion:
 Range
 Inter-quartile Range
 Variance
 Standard Deviation
Relative Measure of Dispersion:
 Coefficient of Variation
Range (R)
The difference between the maximum and
minimum value in a data set, i.e.
R = MAX – MIN
Example: Scores of 15 students in mathematics quiz.
54 58 58 60 62 65 66 71
74 75 77 78 80 82 85
R = 85 - 54 = 31
Some Properties of the Range
The larger the value of the
range, the more dispersed the
observations are.
 It is quick and easy to
understand.
 A rough measure of dispersion.

Inter-Quartile Range (IQR)
The difference between the third quartile and
first quartile, i.e.
IQR = Q3 – Q1
Example: Scores of 15 students in mathematics quiz.
54
74
58
75
58 60 62 65 66 71
77 79 82 82 85
IQR = 78 - 61 = 17
Some Properties of IQR
 Reduces the influence of extreme
values.
 Not as easy to calculate as the
Range.
 Consider only the middle 50% of
the scores in the distribution
Quartile deviation (QD)
is based on the range of the
middle 50% of the scores, instead
of the range of the entire set.
it indicates the distance we need
to go above and below the median
to include approximately the
middle 50% of the scores.
Variance
 important measure of variation
 shows variation about the mean
Population variance
Sample variance
Standard Deviation (SD)
 most important measure of variation
 square root of Variance
has the same units as the original data
 is the average of the degree to which a
set of scores deviate from the mean
value
it is the most stable measures of
variability
Population SD
Sample SD
Computation of Standard Deviation
Data: 10
12 14 15 17
students in mathematics quiz.
n=8
18
18
24 are the scores of
Mean =16
2
2
2
2
2
2
2
2
(10 16)  (12 16)  (14 16)  (15 16)  (17 16)  (18 16)  (18 16)  (24 16)
s
7
 4.309
Remarks on Standard Deviation
 If there is a large amount of variation, then
on average, the data values will be far from
the mean. Hence, the SD will be large.
 If there is only a small amount of variation,
then on average, the data values will be
close to the mean. Hence, the SD will be
small.
Comparing Standard Deviation
Section A
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 3.338
Section B
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = .9258
Section C
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
s = 4.57
Comparing Standard Deviation
Example: Team A - Heights of five marathon players in inches
65”
Mean = 65
S
=0
65 “
65 “
65 “
65 “
65 “
Comparing Standard Deviation
Example: Team B - Heights of five marathon players in inches
Mean = 65”
s = 4.0”
62 “
67 “
66 “
70 “
60 “
Properties of Standard Deviation
 It is the most widely used measure of
dispersion. (Chebychev’s Inequality)
 It is based on all the items and is rigidly
defined.
 It is used to test the reliability of measures
calculated from samples.
 The standard deviation is sensitive to the
presence of extreme values.
 It is not easy to calculate by hand (unlike
the range).
Chebyshev’s Rule
 It permits us to make statements about the
percentage of observations that must be within
a specified number of standard deviation from
the mean
 The proportion of any distribution that lies
within k standard deviations of the mean is at
least 1-(1/k2) where k is any positive number
larger than 1.
 This rule applies to any distribution.
Chebyshev’s Rule
 For any data set with mean () and standard
deviation (SD), the following statements apply:
 At least 75% of the observations are within
2SD of its mean.
 At least 88.9% of the observations are within
3SD of its mean.
Illustration
At least 75%
At least 75% of the observations
are within 2SD of its mean.
Example
The pre-test scores of the 125 LET reviewees last year had
a mean of 70 and a standard deviation of 7 points.
Applying the Chebyshev’s Rule, we can say that:
1. At least 75% of the students had scores between 56
and 84.
2. At least 88.9% of the students had scores between 49
and 91.
Coefficient of Variation (CV)
 measure of relative variation
 usually expressed in percent
 shows variation relative to mean
 used to compare 2 or more groups
 Formula :
 SD 
CV  
  100%
 Mean 
Comparing CVs
 Group A: Average Score = 90
SD = 15
CV = 16.67%
 Group B: Average Score = 92
SD = 10
CV = 10.86%
Example
Mark Erick is one-half standard
deviation above the mean of his group
in math and one standard deviation
above English. What does this imply?
a. He excels in both English and Math.
b. He is better in Math than English.
c. He does not excel in English nor in
Math.
d. He is better is English than Math.
Example
Which statement about the standard
deviation is CORRECT?
a. The lower the standard deviation the
more spread the scores are.
b. The higher the standard deviation the
less the scores spread
c. The higher the standard deviation the
more the spread the scores are
d. It is a measure of central tendency
Example
Which group of scores is most
varied? The group with________.
a. sd = 9
b. sd = 5
c. sd = 1
d. sd = 7
Example
Mean is to Measure of Central
Tendency as___________ is to
measure of variability.
a. Quartile Deviation
b. Quartile
c. Correlation
d. Skewness
Example
HERE ARE TWO SETS OF SCORES:
SET A : 1,2,3,4,5,6,7,8,9
SET B : 3,4,4,5,5,6,6,7,9
Which statement correctly applies to the two
sets of score distribution?
a. The scores in Set A are more spread out than those in
set B.
b. The range for Set B is 5.
c. The range for Set A is 8.
d. The scores in Set B are more spread out than those in
Set A.
Measure of Skewness
 Describes the degree of departures of the

distribution of the data from symmetry.
The degree of skewness is measured by the
coefficient of skewness, denoted as SK and
computed as,
3Mean  Median
SK 
SD
What is Symmetry?
A distribution is said to be symmetric about the mean, if the
distribution to the left of mean is the “mirror image” of the
distribution to the right of the mean. Likewise, a symmetric
distribution has SK=0 since its mean is equal to its median and its
mode.
Measure of Skewness
SK > 0
positively skewed
SK < 0
negatively skewed
Areas Under the Normal Curve
Correlation
refers to the extent to which the
distributions are related or
associated.
the extent of correlation is
indicated by the numerically by the
coefficient of correlation.
the coefficient of correlation
ranges from -1 to +1.
Types of Correlation
1. Positive Correlation
a) High scores in distribution A are
associated with high scores in
distribution B.
b) Low scores in distribution A are
associated with low scores in
distribution B.
2. Negative Correlation
a) High scores in distribution A are
associated with low scores in
distribution B.
b) Low scores in distribution A are
associated with high scores in
distribution B.
3. Zero Correlation
a) No association between distribution A and
distribution B. No discernable pattern.
Positive Correlation
Science Score
30
25
20
15
10
5
0
0
5
Math Score
10
15
20
25
Science Score
Math Score
Negative Correlation
Science
Math
No Correlation
Example
Skewed score distribution means:
a. The scores are normally distributed.
b. The mean and the median are equal.
c. Consist of academically poor
students.
d. The scores are concentrated more at
one end or the other end
Example
Skewed score distribution means:
a. The scores are normally distributed.
b. The mean and the median are equal.
c. Consist of academically poor
students.
d. The scores are concentrated more at
one end or the other end
Example
What would be most likely most the
distribution if a class is composed of
bright students?
a. platykurtic
b. skewed to the right
c. skewed to the left
d. very normal
Example
All the students who took the examination,
got
scores above the mean. What is the
graphical
representation of the score distribution?
a. normal curve
b. mesokurtic
c. positively skewed
d. negatively skewed
A class is composed of
academically poor students. The
distribution most likely to
be______________.
a. skewed to the right
b. a bell curve
c. leptokurtic
d. skewed to the left
Z-SCORE
In statistics, a standard score (also called zscore) is a dimensionless quantity derived by
subtracting the population mean from an
individual (raw) score and then dividing the
difference by the population standard deviation.

The Z-score reveals how many units of the
standard deviation a case is above or below the
mean. The z-score allows us to compare the
results of different normal distributions,
something done frequently in research.

The Standard score is :
where
X is a raw score to be standardized
σ is the standard deviation of the
population
µ is the mean of the population
The quantity z represents the distance between
the raw score and the population mean in units
of the standard deviation. z is negative when the
raw score is below the mean, positive when
above.
A key point is that calculating z requires
the population mean and the population
standard deviation, not the sample mean or
sample deviation. It requires knowing the
population parameters, not the statistics of a
sample drawn from the population of interest.
N) T-SCORE
it is equivalent to ten times the Z-score
plus fifty
T=10Z + 50
EXAMPLE: Based on the table shown, who
performed better, JR or JM? Assume a normal
distribution.
Student
JR
JM
Raw Score
75
58
Mean
65
52
Standard Deviation
4
2
For JR
For JM
JM performed better than JR due to a greater value of z.
From the previous example, the T-score of JR is
T JR = 10(2.5) + 50 = 75
While the T-score of JM is
T JM = 10(3) + 50 = 80
Therefore, JM performed better than JR due to
higher T-score
O) STANINE
Stanine (Standard NINE)
Is a method of scaling test scores on a
nine-point standard scale in a normal distribution.
Percentage
Distribution
4%
7%
12%
17%
20%
17%
12%
7%
4%
Cumulative
Percentage
Distribution
4%
11%
23%
40%
60%
77%
89%
96%
100%
STANINE
1
2
3
4
5
6
7
8
9
Example
Study this group of test which was administered to a
class to whom Jar-R belongs, then answer the
question:
Subject
Math
Physics
English
PE
Mean
56
55
80
75
SD
Jay-R’s Score
10
43
9.5
51
11.25
88
9.75
82
In which subject (s) did Jay-R
perform most poorly in relation to
the group’s mean performance?
A. English
B. Physics
C. PE
D. Math
Based on the data given , what type
of learner is Jay-R?
A. Logical
B. Spatial
C. Linguistic
D. Bodily-Kinesthetic
Based on the data given , in which
subject (s) were scores most
widespread?
A. Math
B. Physics
C. PE
D. English
References
Pilot Training Course on Teaching Basic Statistics by
Statistical Research and Training Center Philippine
Statistical Association , Inc. (Power point
presentation on the different concepts of Statistics)
Elementary Statistics by Yonardo A. Gabuyo et. al.
Rex Book Store
Assessment of Learning I and II by Dr. Rosita De
Guzman-Santos, LORIMAR Publishing, 2007 Ed.
Measurement and Evaluation Concepts and
Principles by Abubakar S. Asaad and Wilham M.
Hailaya, Rex Book Store
LET Reviewer by Yonardo A. Gabuyo, MET Review
Center